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Can you cover this board: |
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with six copies of this tile? | 
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Can you cover this board: |
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with nine copies of this tile? | 
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Solution:
We can do the first one but not the second one. Here is one solution to the first one.
The second one needs an argument something like this. The top left hand square has to be covered. This can only be done if the tile is aligned with one of the sides. Because of the symmetry of the square, both of these ways are the same. So we can assume that the tile is as shown in this picture.
But this leaves two squares on the top right hand side of the board. These can only be filled by using two tiles oriented as shown.
But now the bottom right hand corner squares can only be covered by using tiles as shown.
But now the same argument applies to the bottom left hand squares so we can add two more tiles as shown.
You should now see that we can only add one more tile and this will leave a 2 × 2 gap in the middle of the 6 × 6 board. As all of these moves were forced, we cannot tile the 6 × 6 board with the 1 × 4 shape.